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Design discrete linear-quadratic (LQ) regulator for continuous plant

`lqrd `

[Kd,S,e] = lqrd(A,B,Q,R,Ts)

[Kd,S,e] = lqrd(A,B,Q,R,N,Ts)

`lqrd `

designs a discrete
full-state-feedback regulator that has response characteristics similar
to a continuous state-feedback regulator designed using `lqr`

.
This command is useful to design a gain matrix for digital implementation
after a satisfactory continuous state-feedback gain has been designed.

`[Kd,S,e] = lqrd(A,B,Q,R,Ts) `

calculates the discrete state-feedback law

$$u[n]=-{K}_{d}x[n]$$

that minimizes a discrete cost function equivalent to the continuous cost function

$$J={\displaystyle {\int}_{0}^{\infty}\left({x}^{T}Qx+{u}^{T}Ru\right)}dt$$

The matrices `A`

and `B`

specify
the continuous plant dynamics

$$\dot{x}=Ax+Bu$$

and `Ts`

specifies the sample time of the discrete
regulator. Also returned are the solution `S`

of
the discrete Riccati equation for the discretized problem and the
discrete closed-loop eigenvalues` e = eig(Ad-Bd*Kd)`

.

`[Kd,S,e] = lqrd(A,B,Q,R,N,Ts) `

solves the more general problem with a cross-coupling term in the
cost function.

$$J={\displaystyle {\int}_{0}^{\infty}\left({x}^{T}Qx+{u}^{T}Ru+2{x}^{T}Nu\right)dt}$$

The discretized problem data should meet the requirements for `dlqr`

.

The equivalent discrete gain matrix `Kd`

is
determined by discretizing the continuous plant and weighting matrices
using the sample time `Ts`

and the zero-order hold
approximation.

With the notation

$$\begin{array}{cc}\Phi (\tau )={e}^{A\tau},& {A}_{d}=\Phi ({T}_{s})\\ \Gamma (\tau )={\displaystyle {\int}_{0}^{\tau}{e}^{A\eta}Bd\eta ,}& {B}_{d}=\Gamma ({T}_{s})\end{array}$$

the discretized plant has equations

$$x[n+1]={A}_{d}x[n]+{B}_{d}u[n]$$

and the weighting matrices for the equivalent discrete cost function are

$$\left[\begin{array}{cc}{Q}_{d}& {N}_{d}\\ {N}_{d}^{T}& {R}_{d}\end{array}\right]={\displaystyle {\int}_{0}^{{T}_{s}}\left[\begin{array}{cc}{\Phi}^{T}\left(\tau \right)& 0\\ {\Gamma}^{T}\left(\tau \right)& I\end{array}\right]}\left[\begin{array}{cc}Q& N\\ {N}^{T}& R\end{array}\right]\left[\begin{array}{cc}\Phi \left(\tau \right)& \Gamma \left(\tau \right)\\ 0& I\end{array}\right]d\tau $$

The integrals are computed using matrix exponential formulas
due to Van Loan (see [2]). The
plant is discretized using `c2d`

and the gain matrix
is computed from the discretized data using `dlqr`

.

[1] Franklin, G.F., J.D. Powell, and M.L.
Workman, *Digital Control of Dynamic Systems*,
Second Edition, Addison-Wesley, 1980, pp. 439-440.

[2] Van Loan, C.F., "Computing Integrals Involving
the Matrix Exponential," *IEEE ^{®} Trans. Automatic Control*,
AC-23, June 1978.